Between 1799 and 1804 Gauss had been trying to move ahead on another path. Notes dated 1803 show several attempts by means of geometric constructions and functional equations derived from them (the same process he had applied to the area of a triangle) to develop the relations between the parts of a triangle. But these efforts were in vain. There was still some doubt in his mind. As late as 1808 he considered his work on the parallel axiom unfinished and was not yet fully convinced that it could not be proved.
The following remark by Gauss in 1813 throws some light on his views at the time: “In the theory of parallel lines we are now no further than Euclid was. This is the partie honteuse of mathematics, which sooner or later must get a very different form.” (Collected Works, VIII, 166.)
In 1816 Gauss published a book review of two attempts at proof of the axiom. He expressed a different tone, and spoke of the “vain effort to conceal with an untenable tissue of pseudo proofs the gap which one cannot fill out.” He wanted to indicate his conviction of the impossibility of proving the eleventh axiom.
By 1816 Gauss was in possession of non-Euclidean trigonometry. His pupil Friedrich Ludwig Wachter (1792–1817), who was a professor in Danzig, visited him in Göttingen. Incidentally, Gauss had a high opinion of Wachter’s talent and was deeply distressed in 1817 when Wachter disappeared and was never found.46 Wachter had reviewed in 1816 one of the two works on parallel theory which Gauss had reviewed. During his visit with Gauss in April, 1816, their conversation turned to the foundations of geometry. Gauss stimulated Wachter to do some research on what he called “anti-Euclidean” geometry. Gauss had talked to Wachter about his “transcendental” trigonometry, and Wachter had made vain efforts to penetrate it. In a letter to Gerling on March 16, 1819, Gauss wrote that he had developed non-Euclidean geometry to such an extent that he could completely solve all problems, as soon as the constant = C was given. On April 28, 1817, he wrote Olbers:
I am coming more and more to the conviction that the necessity of our geometry cannot be proved, at least not by human intelligence nor for human intelligence. Perhaps we shall arrive in another existence at other insights into the essence of space, which are now unattainable for us. Until then one would have to rank geometry not with arithmetic, which stands a priori, but approximately with mechanics.
It is not known in what way Gauss arrived at non-Euclidean trigonometry. In his papers is the development of formulas which was probably written down in 1846. He used the process of geometric constructions and of functional equations derived from them, a process which he had used without success in 1803. It is probable that between 1813 and 1816 he advanced on this path.
His hints about the impossibility of proving the parallel axiom, contained in his book review of 1816, did not meet the success Gauss expected, in fact they were criticized, and he decided not to publish his views during his lifetime. He was soon surprised and made happy to find two others moving in the same direction in which he was traveling.
The jurist F. C. Schweikart (1780–1859) had published a work on parallel theory in 1807 in which he objected to the introduction of the infinite in the usual explanation of parallels as non-intersecting straight lines, and demanded that in building up geometry one should proceed from the existence of squares. Later, between 1812 and 1816, without the help of the eleventh Euclidean axiom, he had developed a geometry which he called “astral geometry.” In 1816 he was called from Charkov to the University of Marburg, and in 1818 discussed his system with his colleague Gerling. On January 25, 1819, Gerling wrote thus to Gauss:
I told him how you had expressed yourself publicly several years ago (1816), that fundamentally one had not advanced here since Euclid’s time; indeed, that you had told me several times how you through manifold occupation with this subject had not come to a proof of the absurdity of such an assumption [of a non-Euclidean geometry].
Schweikart asked Gerling to forward a short sketch of his “theory of astral magnitudes” to Gauss and request his opinion. In his reply Gauss stated that almost all of it was as though it had come from his own pen. He found his concept that space is a reality outside of us, whose laws we cannot completely prescribe and whose properties rather are to be completely ascertained only on the basis of experience. The word “astral” chosen by Schweikart was supposed to express the fact that in measurements of magnitude, as they occur in the celestial world, deviations from Euclidean geometry could be observed. It seems to have pleased Gauss, for in later notes he used the word.
A nephew of Schweikart, also a jurist, F. A. Taurinus (1794–1874), had as a young man busied himself with the parallel theory. He was definitely stimulated by his uncle’s book. In October, 1824, he sent an attempted proof to Gauss, who knew since 1821 through the uncle that Taurinus was investigating the foundations of geometry, and recognized in the young man “a thinking mathematical head.”
In a long letter dated November 8, 1824, Gauss replied to Taurinus. He fully explained his views on the parallel axiom, but enjoined the recipient of the letter to make no public use of this private communication or any use that could lead to publicity. Gauss wrote:
The assumption that the sum of the three angles (of a triangle) is less than 180° leads to a special geometry, quite different from ours (Euclidean), which is absolutely consistent and which I have developed quite satisfactorily for myself, so that I can solve every problem in it, with the exception of the determination of a constant which cannot be found out a priori. The larger one assumes this constant, the closer one approaches Euclidean geometry and an infinitely large value makes the two coincide. If non-Euclidean geometry were the true one, and that constant in some relationship to such magnitudes as are in the domain of our measurements on earth or in the heavens, then it could be found out a posteriori.
This letter stimulated young Taurinus to continue his research with increased zeal. In 1825 he published his Theorie der Parallellinien in which he was convinced of the unconditional validity of the parallel axiom, but he began to develop the results which are yielded by a rejection of it. Thus he arrived at that constant which would be peculiar to non-Euclidean geometry. In the simultaneous possibility of infinitely many such geometries, each of which is without internal contradiction, he saw sufficient reason to reject them all.
A copy of the book was sent to Gauss, who also received a copy of Taurinus’ Geometriae prima elementa (1826). Gauss did not send the author an acknowledgment of either one; probably he was offended because Taurinus had mentioned him in the preface of each work. In the second book the author developed the formulas of non-Euclidean trigonometry with one stroke by making the radius of the sphere imaginary in the corresponding formulas of spherical trigonometry. He applied these formulas to the solution of a series of problems and calculated correctly the circumference and area of a circle and the surface and volume of a sphere.
The work of Taurinus was forgotten for many decades. In a letter to Gauss dated December 29, 1829, he wrote: “The success proved to me that your authority is needed to furnish recognition for them (his works), and this first literary attempt, instead of recommending me, as I had hoped, has become for me a rich source of dissatisfaction.”
Letters of Gauss show that in late 1827 he began to do some intense research on the foundations of geometry—he called it “the metaphysics of space doctrine.” In 1828 he wrote that he had first studied these foundations forty years previously and that he would probably not live long enough to work out for publication the full results. The main reason was that he feared what he called “the clamor of the Boeotians” if he were to speak out fully on so revolutionary a matter.
In a short note dated November, 1828, Gauss proved, independently of the eleventh axiom, that the sum of the angles of a triangle cannot be greater than two right angles. In April, 1831, he had begun to write down some of this research because he did not want it to perish with him. Some of these notes have been published in Volume VIII of the Collected Works (pp. 202�
��209). In them the fundamental properties of parallels or “asymptotic straight lines,” as Bolyai called them, are deduced. Gauss arrived in the last note at the paracycle, the curve which results when the radius of a circle becomes infinite. He calls it “trope” (cercle tropique), a clear sign that he conceived the paracycle as the transition from actual circles to hypercycles. A letter from Gauss to Schumacher dated July 12, 1831, discussed the results of the abolition of similarity in non-Euclidean geometry and indicated the formulas valid there for the circumference of a circle.
On November 3, 1823, Johann Bolyai wrote his father that he had “created a new, different world out of nothing.” In February, 1825, he presented to the elder Bolyai the first sketch of his work. The father was not in agreement with it; he was annoyed by the occurrence of the indefinite constants and the many hypothetical systems thereby rendered possible. Father and son could not agree, and finally Johann decided to compose the essence of the work in Latin, attach it to Wolfgang’s planned Tentamen, and send it to Gauss for his opinion.
Reprints of the Appendix scientiam spatii absolute veram exhibens were ready in June, 1831, and one of them was sent to Gauss. Owing to the cholera epidemic. Gauss received only the covering letter, at the close of which Johann wrote a brief outline of his work. After a long time the reprint was returned to Wolfgang. It finally reached Gauss in February, 1832, through a friend of the Bolyais, Baron von Zeyk, who was studying in Göttingen.
The first impression on Gauss was favorable. He found in the book all his own ideas and results developed with great elegance, although written in a form difficult to follow for one who was not familiar with such research, because of the great concentration demanded. Gauss recognized that his own ideas of 1798 on the subject were less mature and developed that those of Johann. He considered young Bolyai “a genius of the first magnitude.”
Gauss wrote Wolfgang on March 6, 1832, expressing surprise at the close agreement of Johann’s results with his own and sent the young geometer hearty greetings along with assurances of his especial high esteem. Wolfgang wrote his son: “Gauss’ answer respecting your work is very fine and redounds to the honor of our fatherland and nation. A good friend says it would be a great satisfaction.”
Johann was greatly disappointed, insulted, and embittered because Gauss gave no public recognition to the Appendix and claimed priority of discovery. In later years relations between him and his father were strained, probably due in large measure to this matter.
In the above-mentioned letter Gauss gave as a sample of his own research a simple proof of the theorem that in non-Euclidean geometry the area of a triangle is proportional to the deviation of the sum of the angles from 180 degrees. It is certain that this represents a part of his research of September, 1799, since the receipt of Johann’s Appendix must have evoked in him memories of his friendship with the elder Bolyai at that time. In the same letter Gauss urged Johann to busy himself with the corresponding problem for space, namely, to determine the cubic content of the tetrahedron (space bounded by four planes). The papers of Johann contain several processes which can serve as a solution, among them the method which Gauss had in mind and which he indicated in one of his notebooks at the time of sending his letter to Wolfgang on March 6, 1832.
A second note by Gauss, dated about 1841, refers to the volume of the tetrahedron. It is on a piece of paper which was found between the leaves of a reprint of the memoir (1836) of the Russian mathematician Nikolai Ivanovitch Lobachevsky (1793–1856) on the application of imaginary geometry to several integrals. By the term “imaginary” geometry he meant non-Euclidean geometry.
As late as 1815–1816 in lectures he gave at the University of Kasan, Lobachevsky was still in the Euclidean fold and had made several attempts to prove the parallel axiom. In the years immediately following he dared to explore the consequences of assuming that the axiom does not exist, and gradually familiarized himself with the thought of the impossibility of proving it. He supported this viewpoint in an unpublished textbook on geometry (1823). Soon thereafter he arrived at the recognition that there is a noncontradictory geometry which does not need the axiom of parallels. He developed this geometry so far that he could treat all its problems purely analytically and gave general rules for the calculations of lengths of arc, areas of surfaces, and volumes. The results of these researches were presented to the Kasan scientific society on February 12, 1826, but were not published until 1829 and 1830 in the Kasan Messenger. A series of additional memoirs followed them in the years 1835–1838.
In order to disseminate his ideas in western Europe, Lobachevsky published in 1837 in Crelle’s Journal47 a short résumé of his imaginary geometry; unfortunately it was poorly adapted as an introduction to the subject. This article seems to have escaped the attention of Gauss, who probably never heard of Lobachevsky until 1840, when he read in Gersdorf’s Repertorium der gesammten deutschen Literatur48 an unfavorable review of the German version of Lobachevsky’s Geometrische Untersuchungen zur Theorie der Parallellinien. Gauss considered the review rather silly.
About the same time (1840) Ernst Knorr (1805–1879), a physicist at the University of Kasan and a friend of Lobachevsky’s, visited Gauss and gave him a copy of Lobachevsky’s above-mentioned memoir of 1846. Later Gauss’ friend Friedrich Georg Wilhelm Struve (1793–1864), director of the observatory at Pulkova, sent him the other memoirs by Lobachevsky which had appeared in the Kasan learned society journal. It is not known where Gauss got the memoir in the Kasan Messenger of 1829–1830.
Fortunately Gauss was able to read the works of Lobachevsky in the original. In a letter to Gerling on February 8, 1844, he compared the short memoirs to “a confused forest through which it is difficult to find a passage and perspective, without having first gotten acquainted with all the trees individually.” On the other hand he praised the conciseness and precision of the Geometrische Untersuchungen. He repeated this praise in a letter to Schumacher dated November 28, 1846, and stated that Lobachevsky had taken a path different from his own, but in a masterly fashion and in genuinely geometric spirit. He got “exquisite enjoyment” from reading it.
On a piece of paper found in one of the two copies49 of Lobachevsky’s Geometrische Untersuchungen belonging to Gauss is the outline of a deduction of formulas of non-Euclidean trigonometry. Probably he wrote it in 1846 when he had occasion to look through the work again. The note gives Gauss’ results of 1816, but a concept is added which is of a later date. As a final result formulas are derived which are identical with the equations of spherical trigonometry, referred to a sphere of radius 1/k; the corresponding equations of non-Euclidean trigonometry follow from those if a purely imaginary value is given to the constant k. Lobachevsky had noted this relationship at the close of the Geometrische Untersuchungen. It occurs there as a special case. It is not known whether Gauss desired to indicate by the letter k that the two geometries can be subordinated to the more general concept of the geometry of a manifold of constant measure of curvature. In the last mentioned letter to Schumacher he wrote: “You know that I have had the same conviction for 54 years (since 1792), with a certain later extension which I do not want to mention here.”
It is not known just what this “later extension” was. Possibly he granted full equality to the geometries yielded by the sign of the measure of curvature. His pupil Riemann later developed the thought that one need only conceive space as an unlimited not as an infinite manifold.
On November 23, 1842, Gauss proposed Lobachevsky as a corresponding member of the Royal Society of Sciences in Göttingen, with the citation that he was one of the most distinguished mathematicians of the Russian Empire, and he was immediately elected. Gauss wrote in his own hand a letter to accompany the diploma of membership. Lobachevsky was highly pleased to get this recognition from a foreign country and especially from the hands of a man he had learned to admire in his youth. His cordial letter of thanks to Gauss was dated June, 1843, in which he excused the tardiness
of his reply by reference to the burning of the city, which had affected his health and personal affairs and overburdened him with official business.
Why did Gauss never mention Bolyai and Lobachevsky in his printed works? This question has often been asked, and Gauss has been criticized for it. The only answer is that he had firmly resolved never to publish anything on parallel theory during his lifetime. And he adhered to this decision. He would do anything short of publication to aid and encourage others, but he felt he had to avoid controversy. He did not mind if others anticipated him and published results he had had for many years.
Otto Struve visited Gauss for the last time in August, 1843, and found him reading one of the short works of Lobachevsky, which, as he said, interested him on account of their content as well as on account of the Russian language, which he was then eagerly studying. A result of the conversation was that by the end of 1843 Struve had sent Gauss as many of Lobachevsky’s writings as he could find in St. Petersburg. It is interesting to note that Dirichlet as early as 1827 discussed non-Euclidean geometry with Gauss on the occasion of his visit in Göttingen.
In a little book published in 1851 Wolfgang Bolyai praised Lobachevsky’s Geometrische Untersuchungen, which was probably the only work of his that he or his son Johann ever saw. Johann studied the relationship of this work to his Appendix. Lobachevsky had the priority of publication.
When Johann Bolyai reported to his father on November 3, 1823, about his new discoveries, the latter urged him to hasten publication because “many things have an epoch simultaneously when they are found in several places, just as in the spring violets come to light in several places.” We know today how right he was.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 22